Convolution and correlation of nearest-neightbor model in algebraic signal processing

The algebraic structure, as an important mathematical tool, is shown to be one of the most powerful tools for the representation of the classical signal processing methods. The classical signal processing concepts can be represented by a whole frame which is provided by the algebraic signal processing (ASP). In ASP, the signal model is defined as a triple (A,M,Φ), where A is a chosen algebra filters, M is an associated A-module of signals and Φ generalizes the idea of a z-transform. The shift operator is the basis of building signal models. The 1-D nearest-neighbor signal model is one of the significant signal models in ASP which result from the 1-D nearest-neighbor shift operator. This paper derives the convolution and correlation theorems of the 1-D continuous and discrete nearest-neighbor signal model in the Fourier domain, and presents the relationship between the new theorems and classical theories. At last, the new autocorrelation and cross-correlation functions are simulated.